Fermat's Last Theorem
FLT 8-19-11 6-25a There may be a key to Pierre Fermat's conceptualization. Fermat did not assert that there is no'' A or B'' such that there can be a solution. He asserted there is'' no cube'' added to another cube such that there can be a solution. ( p265, Devlin, K., Mathematics the New Golden Age, 1999, p265 .) Fermat's point of departure was A^ N^ and B^'' N^. To work from Fermat's perspective one has to let go of the question, is A or B irrational. The construction here begins by defining A as always rational. The expansion of any rational A always produces a rational A ^'N^'. Pierre Fermat worked with rational numbers, I.e. , with A^'N^+''' B^'N^' = 1,' '''rather than A^'N^+'B^N^'' = C^''N^ ''(Devlin, ibid, p282.) Since B ^N^ =(1- A^N^), then, by substitution A^N^+ B ^N^= ''A^N^+ (1-A ^N^'' )= 1. By constructing this problem with A always rational, which always produces a rational A^'N^ ', then (1- A^''N^) ''is also always rational. For any rational A and for any N>2,'' these three knowns are always present, are always rational, and would always follow the same principles. Since the A's are the only possible rational numbers, the expansion to any A^'N^'s yields the only possible rational quantity. For (1-A^'N^) '''to be a part of a solution it has to have the same quantity as one of the A^'N^ ''s. The Fermat question can be expressed most simply as, for any N>2, can a (1-A^'N^)'' have the same quantity as a'' A^'N^? ''This construction completely avoids the nature of B. There is no need to prove that B is always irrational. One proof Fermat had written was ''2 ^N^-2 gives a number that can be divided by that N , also written 2 ^N^-2 modN (''Stark, H.M., ''An Introduction To Number Theory, 1994, MIT Press''.) 2^ N^-2 is also A ^N^-2 when A = 2 of A ^'N^'+''' (1-A'' ^N^) = 1, The way the equation is constructed makes a difference in exploring the issues of the problem. For instance, using the A ^'N^'+''' (1-A ^N^') = 1, ''organizes and simplifies the information such that it is easier to observe essentials, possibilities and alternatives. Constructing an approach that begins with a rational A, a rational A ^'N^', and' a rational (1- A'' ^N^),'' then for any N>2 these three knowns are always present, are always rational, and would always follow the same principles. For all rational sequences of ''A (0,1), i.e.,{.1,.9}, ... {.000...1,.999...9}, every sequence begins with a “1” as .1, .01, .001, .0001, ... .000...1, and this “1” to any power of N creates an ''A ^N^ ''with a “1” as the only digit. No matter what the A^'N^ '''is, the “1” is the begining of the sequence. The next in the sequence is a “2”, as .2, .02, .002, .0002, ... .000...2, and this “2” to any power of N creates an 'A ^N^ 'with Fermat's '''2 N. P.K Tam, in the Southeast Asian Bul of Mathematics v32 ,'' 2008, p1177-81, presents proofs that Fermat's Last Theorem can be understood in “a self-contained elementary and purely algebraic treatment” (p1177). This article has not been discussed in the West. His paper supports the present thesis, which is a “self-contained elementary and purely algebraic treatment.” Another paper ignored is E. E. Escultura's “An Exact Solution of Fermat's Equation ” ''Non-Linear Studies v5, 1998, p227-255, in which he demonstrates that “the loss of certainty affects all concepts and propositions involving... infinite spaces.” The present approach constructs this problem with A always rational, which always produces a rational A^'N^ ', then (1- A^''N^) ''is also always rational. For any rational A and for any N>2,'' these three knowns are always present, are always rational, and would always follow the same principles--and never involves “infinite spaces.” This paper is on the side of Tam, presenting a simple construction that avoids the pitfalls of infinite spaces and has no constraints due to “loss of certainty.” Escultura's telling criticism contributes to a construction avoiding “pitfalls of infinite spaces.''”